black and white hair texture to colored hair
I assume that you already made a simple cycle material and UV wrapped your hair texture on your polygon hair.

Transparency

Since your picture is full black and white you can use it as an alpha mask to mix between your hair color and a transparent shader:

Add --> Shader--> Mix Shader

Add --> Shader--> Transparent BSDF

For the mix shader 


Factor : your picture in an Image Texture
First Shader input : BSDF of the Transparent BSDF node, 
Second Shader input : your color Shader, for the example I simply use a diffuse shader but you can make more complex shader to have a better result
Output of the Mix Shader to the Sufrace of the Material Output Node


Apply a texture

Now you can add a texture to your hair. 

You can add pretty much any kind of color input, in my example case I used just a noise Texture and a color ramp as the input color of a Diffuse Shader, but you can use an Image Texture or any other input color.

Here is the node setup :